Abstract
Introduction/purpose: The running of the coupling constant in various Quantum Field Theories and a possible behaviour of the beta function are illustrated. Methods: The Callan-Symanzik equation is used for the study of the beta function evolution. Results: Different behaviours of the coupling constant for high energies are observed for different theories. The phenomenon of asymptotic freedom is of particular interest. Conclusions: Quantum Electrodynamics (QED) and Quantum Chromodinamics (QCD) coupling constants have completely different behaviours in the regime of high energies. While the first one diverges for finite energies, the latter one tends to zero as energy increases. This QCD phenomenon is called asymptotic freedom.
Highlights
Introduction/purpose: The running of the coupling constant in various Quantum Field Theories and a possible behaviour of the beta function are illustrated
The Callan–Symanzik equation is used for the study of the beta function evolution
Different behaviours of the coupling constant for high energies are observed for different theories
Summary
In (Fabiano, 2021) we have seen how a generic coupling constant behaves at different renormalisation scales. Dt which is a differential equation governing the behaviour of the coupling constant g upon the energy scale considered. As such it needs some initial conditions in order to be solved – a Cauchy problem. For the points 0 and g2 it is clear that the inverse of the previous argument holds true: the coupling g “escapes” from them as t → +∞, and approaches them as energy decreases, for t → 0. Such points are named the infrared stable fixed points.
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